Metamath Proof Explorer


Theorem cplgr2v

Description: An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by AV, 3-Nov-2020)

Ref Expression
Hypotheses cplgr0v.v 𝑉 = ( Vtx ‘ 𝐺 )
cplgr2v.e 𝐸 = ( Edg ‘ 𝐺 )
Assertion cplgr2v ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝐺 ∈ ComplGraph ↔ 𝑉𝐸 ) )

Proof

Step Hyp Ref Expression
1 cplgr0v.v 𝑉 = ( Vtx ‘ 𝐺 )
2 cplgr2v.e 𝐸 = ( Edg ‘ 𝐺 )
3 1 iscplgr ( 𝐺 ∈ UHGraph → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
4 3 adantr ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
5 1 2 uvtx2vtx1edgb ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝑉𝐸 ↔ ∀ 𝑣𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
6 4 5 bitr4d ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝐺 ∈ ComplGraph ↔ 𝑉𝐸 ) )